Reduction Types of Genus-3 Curves in a Special Stratum of their Moduli Space

Bouw, Irene I.; Coppola, Nirvana; Kılıçer, Pınar; Kunzweiler, Sabrina; García, Elisa Lorenzo; Somoza, Anna

doi:10.1007/978-3-030-77700-5_4

Cited by 5 publications

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References 20 publications

(14 reference statements)

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“…We would like to find the minimal regular model Y min of Y in the case where Y has semistable reduction. Recently there has been a flurry of activity finding explicit descriptions for models of different classes of curves, including [8], [9], [15], [14] (hyperelliptic curves), [6] (superelliptic), [11], [5] (genus 3), [7] (∆ v -regular).…”

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Models of Bihyperelliptic Curves

Faraggi

2021

Preprint

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We give an explicit description of the minimal regular model of bihyperelliptic curves with semistable reduction over a local field of odd residue characteristic. We do this using a generalisation of the cluster picture; a completely combinatorial object attached to a hyperelliptic curve y 2 = f (x) over K which contains the data of the p-adic distances between the roots of f . We add some information, resulting in a chromatic cluster picture, and show that this determines the minimal regular model of Y with the action of Frobenius.

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confidence: 99%

Models of Bihyperelliptic Curves

Faraggi

2021

Preprint

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Reduction Types of Genus-3 Curves in a Special Stratum of their Moduli Space

Bouw

Coppola

Kılıçer

et al. 2021

Association for Women in Mathematics Series

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We study a 3-dimensional stratum M 3,V of the moduli space M 3 of curves of genus 3 parameterizing curves Y that admit a certain action of V C 2 × C 2 . We determine the possible types of the stable reduction of these curves to characteristic different from 2. We define invariants for M 3,V and characterize the occurrence of each of the reduction types in terms of them. We also calculate the j-invariant (resp. the Igusa invariants) of the irreducible components of positive genus of the stable reduction Y in terms of the invariants.

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Invariants for trees of non-archimedean polynomials and skeleta of superelliptic curves

Helminck

2022

Math. Z.

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In this paper we generalize the j-invariant criterion for the semistable reduction type of an elliptic curve to superelliptic curves X given by $$y^{n}=f(x)$$ y n = f ( x ) . We first define a set of tropical invariants for f(x) using symmetrized Plücker coordinates and we show that these invariants determine the tree associated to f(x). This tree then completely determines the reduction type of X for n that are not divisible by the residue characteristic. The conditions on the tropical invariants that distinguish between the different types are given by half-spaces as in the elliptic curve case. These half-spaces arise naturally as the moduli spaces of certain Newton polygon configurations. We give a procedure to write down their equations and we illustrate this by giving the half-spaces for polynomials of degree $$d\le {5}$$ d ≤ 5 .

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Reduction Types of Genus-3 Curves in a Special Stratum of their Moduli Space

Cited by 5 publications

References 20 publications

Models of Bihyperelliptic Curves

Models of Bihyperelliptic Curves

Reduction Types of Genus-3 Curves in a Special Stratum of their Moduli Space

Invariants for trees of non-archimedean polynomials and skeleta of superelliptic curves

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